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On a wave equation with supercritical interior and boundary sources and damping terms. (English) Zbl 1244.35092

In this article the following system is investigated: \[ \begin{cases} u_{tt}-\Delta u+g_0(u_t)=|u|^{p-1}u \quad\text{in } \Omega\times [0, \infty),\\ \partial_{\nu} u+u+g(u_t)=|u|^{k-1}u \quad\text{on } \Gamma\times [0, \infty),\\ u(0)=u_0\in H^1(\Omega),\;u_t(0)=u_1\in L^2(\Omega), \end{cases} \tag{1} \] where \(\Omega\subset \mathbb R^3\) is a bounded open set with sufficiently smooth boundary \(\Gamma\), the maps \(g_0(s)\) and \(g(s)\) are monotone and represent the interior and boundary dissipation, the Nemytski operators \(|u|^{p-1}u\), \(p>1\), \(|u|^{k-1}u\), \(k>1\), model the interior and boundary sources respectively.
The authors give conditions for \(g_0\), \(g\) and \(p\), \(k\) under which the problem (1) has unique global weak solution \(u\in {\mathcal C}([0, \infty), H^1(\Omega))\). The exponential and algebraic uniform decay rates of the finite energy are established. The authors prove a blow up result for weak solutions of (1) with nonnegative initial energy.

MSC:

35L71 Second-order semilinear hyperbolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B35 Stability in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B33 Critical exponents in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B44 Blow-up in context of PDEs
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